Models with limited dependent variables

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Models with limited dependent variables. Doctoral Program 2006-2007 Katia Campo. 3. Nested Logit Model. (K.Train, Ch.4, Franses and Paap Ch.5). 3. Nested Logit Model. Choice between J&gt;2 alternatives which can be grouped into subsets based on differences in substitution pattern Example:.

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### Models with limited dependent variables

Doctoral Program 2006-2007

Katia Campo

### 3. Nested Logit Model

(K.Train, Ch.4, Franses and Paap Ch.5)

3. Nested Logit Model
• Choice between J>2 alternatives which can be grouped into subsets based on differences in substitution pattern
• Example:

Choice of transportation mode

Car

Transit

Carpool

Bus

Train

Car alone

3. Nested Logit Model
• Assumption cumulative distribution of error terms
• Probability
3. Nested Logit Model
• Probability can be decomposed into 2 logit models

(1)

(2)

3. Nested Logit Model
• Link between Pni | Bk * Pn Bk (upper and lower model level) through inclusive value Ink
• To comply with RUM k must be in the [0,1] interval
• IIA within, not across nests
3. Nested Logit Model

Estimation

• Joint estimation
• Sequential estimation
• Estimate lower model
• Compute inclusive value
• Estimate upper model with inclusive value as explanatory variable

• Add fourth step, using the parameter estimates as starting values for joint estimation
3. Nested Logit Model
• Example: Location choices by French firms in Eastern and Western Europe
• Dependent var.: probability of choosing location j Pj=P(πj> πk  k≠j )
• Location choices are likely to have a nested structure (non-IIA)
• First, select region (East or Western Europe)
• Next, select country within region

Disdier and Mayer (2004)

3. Nested Logit Model
• Example 1: Location choices by French firms in Eastern and Western Europe

Location choice

E.Eur

W.Eur

………

………

CN

C1

CJ+1

CJ

Disdier and Mayer (2004)

3. Nested Logit Model
• Location choices: Data
• 1843 location decisions in Europe from 1980 to 1999 (official statistics)
• 19 host countries (13 W.Eur, 6 E.Eur)

Disdier and Mayer (2004)

3. Nested Logit Model
• Location choices: Data
• NF French firms already located in the country
• GDP GDP
• GPP/CAP GDP per capita
• DIST Distance France – host country
• W Average wage per capita (manufacturing)
• UNEMPL unemployment rate
• EXCHR Exchange rate volatility
• FREE Free country
• PNFREE Partly free and not free country
• PR1 Country with political rights rated 1
• PR2 Country with political rights rated 2
• PR345 Country with political rights rated 3,4,5
• PR67 Country with political rights rated 6,7
• LI Annual liberalization index
• CLI Cumulative liberalization index
• ASSOC =1 if an association agreement is signed

Disdier and Mayer (2004)

3. Nested Logit Model

Disdier and Mayer (2004)

3. Nested Logit Model

Disdier and Mayer (2004)

3. Nested Logit Model

• Example 2: Shopping centre choice

3. Nested Logit Model

• Example 2: Shopping centre choice

3. Nested Logit Model

• Example 2: Shopping centre choice
3. Nested Logit Model
• (Purchase) incidence and choice models can be linked in the same way

Include incl.value 

Purchase decision

No purchase

Purchase

...

(See above: Bucklin & Gupta)

Altern. J

Altern.1

3. Nested Logit Model
• Other examples
• Private label versus nationals brands
• Same versus other brand
• Fixed rate (low risk) versus variable rate (high risk) investments
• Choice of transportation mode
• ....

### 4. Probit Model

(K.Train, Ch.5 ; Franses and Paap, Ch.4-5)

4. Probit Model
• Based on the general RUM-model
• Ass.: error terms are distributed normal with a mean vector of zero and covariance matrix 
• density function:
4. Probit Model

Logit or Probit?

• Trade-off between tractability and flexibility
• Closed-form expression of the integral for Logit, not for Probit models
• Probit allows for random taste variation, can capture any substitution pattern, allows for correlated error terms and unequal error variances

 Dependent on the specifics of the choice situation

4. Probit Model

Estimation

• Approximation of the multidimensional integral
• Non-simulation procedures (see Kamakura 1989)

Can usually only be applied to restricted cases and/or provide inaccurate estimations

• Simulation procedures (see Geweke et al.1994, Train)
4. Probit Model

Simulation-based estimation(binary probit, CFS)

• Step 1
• For each observation n=1, ..., N draw r ~ N(0,1), (r = 1, ......., R: repetitions)
• Initialize y_count= 0, =mt (starting values)
• Compute y*rn = xn mt + L r ; L= choleski factor (LL’= )
• Evaluate: y*rn >0  y_count= y_count+1
• Repeat R times

Weeks (1997)

4. Probit Model

Simulation-based estimation(binary probit, CFS)

• Step 2: calculate probabilities

Pn| mt= y_count/R

• Step 3: Form the simulated LL function

SLL=  n yn ln(Pn|mt)+(1-yn) ln(1-Pn|mt)

• Step 4: Check convergence criteria (SLL(mt)- SLL(mt-1))
• Step 5: Update mt: mt+1 = mt + v
• Step 6: Iterate (until convergence)

Weeks (1997)

4. Probit Model

Simulation-based estimation: MNProbit

• Based on same principles
• More efficient simulation procedures (see Train)
• Identification: normalization of level and scale
• Re-express model in utility differences
• Normalization of varcov matrix (see Train)
4. Probit Model
• Random taste variation
• Model with random coefficients
• E.g.: n~N(b,W)
• Unj = b’xnj+ *’n xnj + nj
• = b’xnj+ nj
• nj : correlated error terms (dep.on xnj, see Train)
4. Probit Model
• Substitution patterns
• Full covariance matrix (no parameter restrictions) unrestricted substitution patterns
• Structured covariance matrix (restrictions on some covariance parameters)
•  the structure imposed on  determines the substitution pattern and may allow to reduce the number of parameters to be estimated
4. Probit Model
• Example (Kamakura and Srivastava 1984): random utility components ni, nj are more (less) highly correlated when i and j are more (less) similar on important attributes

(dij = weighted eucledian distance between i & j)

4. Probit Model
• Examples
• Choice models at brand-size level: correlation between ≠ sizes of same brand (Chintagunta 1992)

MNL model

gives biased

estimates

of price

elasticity

4. Probit Model
• Examples
• Firm innovation (Harris et al. 2003)
• Binary probit model for innovative status (innovation occurred or not)
• Based on panel data  correlation of innovative status over time: unobserved heterogeneity related to management ability and/or strategy
4. Probit Model
• Examples
• Dynamics of individual health (Contoyannis, Jones and Nigel 2004)
• Binary probit model for health status (healthy or not)
• Survey data for several years  correlation over time (state dependence) + individual-specific (time-invariant) random coefficient
4. Probit Model
• Examples
• Choice of transportation mode (Linardakis and Dellaportas 2003)  Non-IIA substitution patterns

### 5. Ordered Logit Model

5. Ordered Logit Model
• Choice between J>2 ordered ‘alternatives’
• Ordinal dependent variable y = 1, 2, ... J, with

rank(1) < rank(2) < ... < rank(J)

• Example:
• Purchase of 1, 2, ... J units
• Evaluation on a J-point scale ranging from, e.g., ‘dislike very much’ to ‘like very much’
5. Ordered Logit Model
• Suppose yi* is a continous latent variable which
• is a linear function of the explanatory variables

yn* = Xn + n

• and can be ‘mapped’ on an ordered multinomial variable as follows:

yn= 1 if 0 < yn*  1

yn= j if j-1 < yn*  j

yn= J if J-1 < yn*  J

0 < 1 < …. < j < … < J

5. Ordered Logit Model

Ordered logit (see above)

• 0 , J and 0: set equal to zero
5. Ordered Logit Model

Interpretation of parameters (marginal effects)

5. Ordered Logit Model
• Assumption of equal slope k
• Biased estimates if assumption of strictly ordered

outcomes does not hold

• treat outcomes as nonordered unless there are

good reasons for imposing a ranking

5. Ordered Logit Model

Example: Effectiveness of better public transit as a way to reduce automobile congestion and air polution in urban areas

• Research objective: develop and estimate models to measure how public transit affects automobile ownership and miles driven.
• Data: Nationwide Personal Transportation Survey (42.033 hh): socio-demo’s, automobile ownership and use, public transportation avail.

Kim and Kim (2004)

5. Ordered Logit Model
• Dependent variable ownership model = number of cars (k = 0, 1, 2,  3)  ordinal variable
• C*i = latent variable: automobile ownership propensity of hh i
• Relation to observed automobile ownership:

Ci=k if k-1 < ’xi +  < k

• P(Ci=k)=F(k- ’xi) - F(k-1 - ’xi)

Kim and Kim (2004)

5. Ordered Logit Model

• Examples
• Occupational outcome as a function of socio-demographic characteristics (Borooah)
• Unskilled/semiskilled
• Skilled manual/non-manual
• Professional/managerial/technical
• School performance (Sawkins 2002)
• Function of school, teacher and student characteristics
• Level of insurance coverage
D.Heterogeneity
• Observed heterogeneity
• Unobserved heterogeneity
• Over decision makers
• Random coefficients Models
• E.g. Mixed Logit Model (see Train)
• Over segments
• Latent class estimation
D.Heterogeneity: Latent class est.
• Ass.: Consumers can be placed into a small number of – homogeneous - segments which differ in choice behavior ( response parameters)
• Relative size of the segment s (s=1, 2, ..., M) is given by

fs = exp(s) / s’exp(s’)

Kamakura and Russell (1989)

D.Heterogeneity: Latent class est.
• Probability of choosing brand j, conditional on consumer i being a member of segment s

Ps(yn = j|Xn) = exp(Xjns)

lexp(Xlns)

D.Heterogeneity: Latent class est.
• Unconditional probability that consumer i will choose brand j

P(yn = j|Xn) = sfs Ps(yn = j|Xn)

= s exp(s) exp(Xjns)

s’exp(s’) lexp(Xlns)

D.Heterogeneity: Latent class est.
• Estimation: Maximum Likelihood
• Likelihood of a hh’s choice history Hn

L(Hn) = s [ exp(s)L(Hn|s) / s’ exp(s’) ]

with

L(Hn|s) = t Ps(ynt = c(t) | Xnt)

c(t) = index of the chosen option at time t.

• Maximize likelihood over all hh’s: n L(Hn)
D. Heterogeneity: Latent class est.

Segment analysis

• Based on parameter estimates (e.g. difference in price sensitivity)
• Based on segment profiles
• Post-hoc: based on assignment of hh to segments; Probability that hh n belongs to segment s =

P(ns | Hn) = L(Hn|s)fs / s’ [L(Hn|s’)fs’]

Analyze characteristics of different segments

• A priori: make fs a function of variables that may explain segment membership (e.g. income for segments which differ in price sensitivity)
D.Heterogeneity: Latent class est.
• Example: heterogeneity in price sensitivity (Bucklin and Gupta 1992)
D.Heterogeneity: Latent class est.
• Example: heterogeneity in price sensitivity
D.Heterogeneity: Latent class est.

Choice segments: segment 1 = more sensitive to price and promo

Incidence segments: segment 2 and 4 = more sensitive to changes

in category attractiveness (change in price/promo)

 Confirms that  combinations of choice/inc.price sensitivity occur

D. Heterogeneity: Latent class est.
• Segment analysis: price elasticity
D. Heterogeneity: Latent class est.
• Segment analysis: socio-demographic profile